R. L. Eubank, Spline Smoothing and Nonparametric Regression.New York: Marcel Dekker, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a reasonable degrees of freedom parameter. In this approach, a representative 1 D slice of the data is smoothed for all choices of degrees of freedom from 2 to where is the discrete length of the slice. For each choice of degrees of freedom, the spar value is computed (see Appendix B) 17] [20]. A plot of versus degrees of freedom is then generated, from which the final degrees of freedom is selected by finding the tangent point of this graph to a 45 line (Fig. 8) The interpretation of this procedure is that a significant decrease in the degrees of freedom leads to excessive smoothness ....

....method is trial and error in which a value for is chosen and the resulting smooth is observed. If the curve is not sufficiently smooth, the process is repeated. The second technique is by cross validation. The mathematics and theory behind this method are beyond the scope of this paper [17] [20], but it generally achieves a reasonably good estimate for the smoothing parameter. The final technique involves choosing the degrees of freedom. The degrees of freedom is related to the number of knots [20] In this way, the relative smoothness or roughness of a fit can be controlled to in ....

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R. Eubank, Spline Smoothing and Nonparametric Regression. New York: Marcel-Dekker, 1988.

....measurements y. After presenting the maximum likelihood criterion and noting its shortcomings, we introduce a nonparametric optimality criterion. Our approach is motivated by the success of nonparametric regression, especially cubic spline smoothing, at estimating smooth curves from noisy data [43]. A. Maximum Likelihood Criterion An obvious approach would be to find the object set whose computed projections are the closest to the measurements in some sense. Under the independent Gaussian noise model, the least squares estimate is also the maximum likelihood (ML) estimate, defined by (cf. ....

....planes within its length. With these definitions, we can rewrite (10) as: N1 , N K X . 12) The matrix S k , defined in [48, eqn. 23) depends on N k and #, and serves to discretize the integral in (10) Though many desirable properties of spline smoothers are known [43,45], the nonlinearity of (12) limits how much we can say about its theoretical properties. There are probably local minima, and even the global minimum is not unique in general, due to the non uniqueness discussed in Section IV. However, regularization methods have shown promise in other applications ....

R. L. Eubank, Spline smoothing and nonparametric regression. New York: Marcel Dekker, 1988.

....expansion of q. In many cases, only ti is needed since , t, III. SPLINE SMOOTHING OF VECTOR MEASUREMENTS We now generalize the results of the previous section by con sidering the vector measurement model: y = g(t, n = 1, N g(t, t, y, 6 ffl n, e, N(O, 0, n m. 3The objection could be raised that model (I 0) is not as general as model (1) which contains the additional H. term. However, if the lneasummeat matrices H. are all of rank M. then multiplying both sides of (1) by (H, 2.H. HI2,7 transforms (1) into (10) In general, the measurement matrices ....

.... HI2,7 transforms (1) into (10) In general, the measurement matrices may not all be of rank M. If they are not, then even optimal Kalman filters, derived from the state space model (2) will only be effective if the pairs (1t, A. satisfy the technical condillon of stochastic observa bility [23] This condition is usually satisfied because of the presence of delay or difference terms in x, Any such (application dependent) a priori information should be incorporated into the nonparametric paradigm pre sented here. Although we assume the error covariances I; are known for the ....

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R. L. Eubank, Spline Smoothing and Nonparametric Regression. New York: Marcel Dekker, 1988.

....all of the data rather than just data near the estimation point. For global bandwidths, this implies that the estimator is no longer nonparametric (see, for example, Jones, 1993) However, this is not the case for local bandwidths as is discussed below. A second concern is the role of convexity. Eubank (1999) gives an excellent discussion of the di erence between parametric regression and nonparametric regression. Ultimately, parametric regression restricts the mean function to be of a certain parametric form (such as a line) and then uses the data to estimate the parameters of that functional form. ....

Eubank, R.L. (1999), \Nonparametric Regression and Spline Smoothing," New York: Marcel Dekker.

....measurements y. After presenting the maximum likelihood criterion and noting its shortcomings, we introduce a nonparametric optimality criterion. Our approach is motivated by the success of nonparametric regression, especially cubic spline smoothing, at estimating smooth curves from noisy data [43]. A. Maximum Likelihood Criterion An obvious approach would be to find the object set whose computed projections are the closest to the measurements in some sense. Under the independent Gaussian noise model, the least squares estimate is also the maximum likelihood (ML) estimate, defined by (cf. ....

....X1 ; X K N X n=1 kyn Gamma s(x 1 (z n ) xK (z n ) k 2 K X k=1 X 0 k S k X k # : 12) The matrix S k , defined in [48, eqn. 23) depends on N k and ff, and serves to discretize the integral in (10) Though many desirable properties of spline smoothers are known [43,45], the nonlinearity of (12) limits how much we can say about its theoretical properties. There are probably local minima, and even the global minimum is not unique in general, due to the non uniqueness discussed in Section IV. However, regularization methods have shown promise in other applications ....

R. L. Eubank, Spline smoothing and nonparametric regression. New York: Marcel Dekker, 1988.

....of penalized likelihood estimation, which we refer to as smoothing splines methods, and those that use polynomial splines often in conjunction with adaptive model selection. The smoothing spline solution to a function estimation problem is typically the maximizer of a penalized likelihood function[30, 31, 32]. In survival analysis, smoothing splines have been used by Anderson and Senthilselvan[33] Whittemore and Keller[34] Senthilselvan[35] O Sullivan[36, 37] Gray[38] Hastie and Tibshirani[31, 39] and Gu[40, 41] Most of these papers use splines within the framework of the proportional hazards ....

Eubank RL. Spline Smoothing and Nonparametric Regression. New York: Marcel Dekker, 1988.

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R. L. Eubank, Spline Smoothing and Nonparametric Regression.New York: Marcel Dekker, 1988.

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